On a simple method of determing the homology and the cohomology of finitely presented groups

Abstract

In this paper the authors obtained a method of constructing free resolutions of Z for finitely presented groups directly from their presentations by extending Lyndon’s 3-term partial resolution to a full-length resolution. Authors resolutions and the method of their construction are such that free generators of the modules and the boundary homomorphisms are directly and explicitly obtained by solving of linear equations over the corresponding integral group rings, and hence these are immediately applicable for computing homology and cohomology of the groups for arbitrary coefficient modules. Authors have also described a general situation where their method is valid. The method has been used for a number of classes of group including Fuchsian groups, a few Euclidean crystallographic groups, NEC groups, the fundamental groups of a few interesting manifolds, groups of isometries of the hyperbolic plane and a few nilpotent groups of class 2.

Key words: Group presentation; Free resolution; Homology; Cohomology

DOI: 10.3329/jbas.v34i2.6854

Journal of Bangladesh Academy of Sciences, Vol. 34, No. 2, 103-114, 2010